Theory of adaptive subtraction

The goal of adaptive subtraction is as follows: given a time series 322#322 and a desired time series 9#9, we seek a filter 313#313 that minimizes the difference between 323#323 and 9#9 where * is convolution. We can rewrite this definition in the fitting goal  

 324#324 (136)

where 325#325 represents the convolution with the time series 322#322. We can minimize this fitting goal in a least-squares sense leading to the objective function

326#326 (137)

where (') is the transpose. The minimum energy solution is given by  

 327#327 (138)

where 328#328 is the least-squares estimate of 313#313.This approach is very popular but has some intrinsic limitations. In particular 329#329 is by construction orthogonal to the residual 330#330. In the multiple attenuation problem 9#9 is the data, 322#322 the multiple model and 330#330 the estimated primaries. If both signal and noise are correlated, the separation will suffer because of the orthogonality principle.

From now on I will refer to this method as the ``standard approach''.

In the next section I propose improving the adaptive subtraction scheme. This improvement leads to an unbiased matched-filter estimation when both signal and noise are correlated.